bug366 (missed optimization)

Percentage Accurate: 44.2% → 99.2%

Time: 2.7s

Alternatives: 4

Speedup: 111.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (sqrt (+ (* x x) (+ (* y y) (* z z)))))

C

double code(double x, double y, double z) {
	return sqrt(((x * x) + ((y * y) + (z * z))));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = sqrt(((x * x) + ((y * y) + (z * z))))
end function

Java

public static double code(double x, double y, double z) {
	return Math.sqrt(((x * x) + ((y * y) + (z * z))));
}

Python

def code(x, y, z):
	return math.sqrt(((x * x) + ((y * y) + (z * z))))

Julia

function code(x, y, z)
	return sqrt(Float64(Float64(x * x) + Float64(Float64(y * y) + Float64(z * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = sqrt(((x * x) + ((y * y) + (z * z))));
end

Wolfram

code[x_, y_, z_] := N[Sqrt[N[(N[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 44.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (sqrt (+ (* x x) (+ (* y y) (* z z)))))

C

double code(double x, double y, double z) {
	return sqrt(((x * x) + ((y * y) + (z * z))));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = sqrt(((x * x) + ((y * y) + (z * z))))
end function

Java

public static double code(double x, double y, double z) {
	return Math.sqrt(((x * x) + ((y * y) + (z * z))));
}

Python

def code(x, y, z):
	return math.sqrt(((x * x) + ((y * y) + (z * z))))

Julia

function code(x, y, z)
	return sqrt(Float64(Float64(x * x) + Float64(Float64(y * y) + Float64(z * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = sqrt(((x * x) + ((y * y) + (z * z))));
end

Wolfram

code[x_, y_, z_] := N[Sqrt[N[(N[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \mathsf{hypot}\left(z, x\right) \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (hypot z x))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return hypot(z, x);
}

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return Math.hypot(z, x);
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return math.hypot(z, x)

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return hypot(z, x)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = hypot(z, x);
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[Sqrt[z ^ 2 + x ^ 2], $MachinePrecision]

Derivation

1. Initial program 47.0%

   \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]

2. Taylor expanded in y around 0 32.9%

   \[\leadsto \color{blue}{\sqrt{{z}^{2} + {x}^{2}}} \]
3. Step-by-step derivation

   1. unpow2 32.9%

      \[\leadsto \sqrt{\color{blue}{z \cdot z} + {x}^{2}} \]

   2. unpow2 32.9%

      \[\leadsto \sqrt{z \cdot z + \color{blue}{x \cdot x}} \]

   3. hypot-def 70.6%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(z, x\right)} \]

4. Simplified 70.6%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(z, x\right)} \]

5. Final simplification 70.6%

   \[\leadsto \mathsf{hypot}\left(z, x\right) \]

Alternative 2: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{+148}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 1.8e-14)
   (hypot y x)
   (if (<= z 1.26e+148) z (if (<= z 2.3e+163) (hypot y x) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.8e-14) {
		tmp = hypot(y, x);
	} else if (z <= 1.26e+148) {
		tmp = z;
	} else if (z <= 2.3e+163) {
		tmp = hypot(y, x);
	} else {
		tmp = z;
	}
	return tmp;
}

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.8e-14) {
		tmp = Math.hypot(y, x);
	} else if (z <= 1.26e+148) {
		tmp = z;
	} else if (z <= 2.3e+163) {
		tmp = Math.hypot(y, x);
	} else {
		tmp = z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if z <= 1.8e-14:
		tmp = math.hypot(y, x)
	elif z <= 1.26e+148:
		tmp = z
	elif z <= 2.3e+163:
		tmp = math.hypot(y, x)
	else:
		tmp = z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 1.8e-14)
		tmp = hypot(y, x);
	elseif (z <= 1.26e+148)
		tmp = z;
	elseif (z <= 2.3e+163)
		tmp = hypot(y, x);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.8e-14)
		tmp = hypot(y, x);
	elseif (z <= 1.26e+148)
		tmp = z;
	elseif (z <= 2.3e+163)
		tmp = hypot(y, x);
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 1.8e-14], N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision], If[LessEqual[z, 1.26e+148], z, If[LessEqual[z, 2.3e+163], N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision], z]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.7999999999999999e-14 or 1.25999999999999997e148 < z < 2.30000000000000002e163

   1. Initial program 50.4%

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]

   2. Taylor expanded in z around 0 39.1%

      \[\leadsto \color{blue}{\sqrt{{y}^{2} + {x}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 39.1%

         \[\leadsto \sqrt{\color{blue}{y \cdot y} + {x}^{2}} \]

      2. unpow2 39.1%

         \[\leadsto \sqrt{y \cdot y + \color{blue}{x \cdot x}} \]

      3. hypot-def 74.2%

         \[\leadsto \color{blue}{\mathsf{hypot}\left(y, x\right)} \]

   4. Simplified 74.2%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(y, x\right)} \]

   if 1.7999999999999999e-14 < z < 1.25999999999999997e148 or 2.30000000000000002e163 < z

   1. Initial program 37.6%

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]

   2. Taylor expanded in z around inf 67.7%

      \[\leadsto \color{blue}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{+148}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{hypot}\left(y, x\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 3: 77.7% accurate, 13.6× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2.05 \cdot 10^{-16}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+147}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+163}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 2.05e-16)
   (- x)
   (if (<= z 7.8e+147) z (if (<= z 3.5e+163) (- x) z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.05e-16) {
		tmp = -x;
	} else if (z <= 7.8e+147) {
		tmp = z;
	} else if (z <= 3.5e+163) {
		tmp = -x;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2.05d-16) then
        tmp = -x
    else if (z <= 7.8d+147) then
        tmp = z
    else if (z <= 3.5d+163) then
        tmp = -x
    else
        tmp = z
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.05e-16) {
		tmp = -x;
	} else if (z <= 7.8e+147) {
		tmp = z;
	} else if (z <= 3.5e+163) {
		tmp = -x;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if z <= 2.05e-16:
		tmp = -x
	elif z <= 7.8e+147:
		tmp = z
	elif z <= 3.5e+163:
		tmp = -x
	else:
		tmp = z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (z <= 2.05e-16)
		tmp = Float64(-x);
	elseif (z <= 7.8e+147)
		tmp = z;
	elseif (z <= 3.5e+163)
		tmp = Float64(-x);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.05e-16)
		tmp = -x;
	elseif (z <= 7.8e+147)
		tmp = z;
	elseif (z <= 3.5e+163)
		tmp = -x;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 2.05e-16], (-x), If[LessEqual[z, 7.8e+147], z, If[LessEqual[z, 3.5e+163], (-x), z]]]

Derivation

1. Split input into 2 regimes

2. if z < 2.05000000000000003e-16 or 7.80000000000000033e147 < z < 3.5000000000000003e163

   1. Initial program 50.4%

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]

   2. Taylor expanded in x around -inf 20.6%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   3. Step-by-step derivation

      1. mul-1-neg 20.6%

         \[\leadsto \color{blue}{-x} \]

   4. Simplified 20.6%

      \[\leadsto \color{blue}{-x} \]

   if 2.05000000000000003e-16 < z < 7.80000000000000033e147 or 3.5000000000000003e163 < z

   1. Initial program 37.6%

      \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]

   2. Taylor expanded in z around inf 67.7%

      \[\leadsto \color{blue}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.05 \cdot 10^{-16}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+147}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+163}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 4: 51.9% accurate, 111.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ z \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 z)

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return z;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return z;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return z

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return z
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = z;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := z

Derivation

1. Initial program 47.0%

   \[\sqrt{x \cdot x + \left(y \cdot y + z \cdot z\right)} \]

2. Taylor expanded in z around inf 19.5%

   \[\leadsto \color{blue}{z} \]

3. Final simplification 19.5%

   \[\leadsto z \]

Developer target: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{hypot}\left(x, \mathsf{hypot}\left(y, z\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (hypot x (hypot y z)))

C

double code(double x, double y, double z) {
	return hypot(x, hypot(y, z));
}

Java

public static double code(double x, double y, double z) {
	return Math.hypot(x, Math.hypot(y, z));
}

Python

def code(x, y, z):
	return math.hypot(x, math.hypot(y, z))

Julia

function code(x, y, z)
	return hypot(x, hypot(y, z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = hypot(x, hypot(y, z));
end

Wolfram

code[x_, y_, z_] := N[Sqrt[x ^ 2 + N[Sqrt[y ^ 2 + z ^ 2], $MachinePrecision] ^ 2], $MachinePrecision]

Reproduce

herbie shell --seed 2023195 
(FPCore (x y z)
  :name "bug366 (missed optimization)"
  :precision binary64

  :herbie-target
  (hypot x (hypot y z))

  (sqrt (+ (* x x) (+ (* y y) (* z z)))))